Lattice norms on the unitization of a truncated normed Riesz space

TL;DR

This paper investigates lattice norms on the unitization of truncated Riesz spaces, establishing conditions for their extension, and characterizes the extremal lattice norms, linking the structure to Banach lattice properties.

Contribution

It provides necessary and sufficient conditions for extending lattice norms to the unitization of truncated Riesz spaces and characterizes the extremal lattice norms, connecting to Banach lattice theory.

Findings

Existence of a largest and smallest lattice norms extending the given norm

Any alternative lattice norm is either equivalent to the largest or equal to the smallest

Unitization is a Banach lattice if and only if the original space is a Banach lattice

Abstract

Truncated Riesz spaces was first introduced by Fremlin in the context of real-valued functions. An appropriate axiomatization of the concept was given by Ball. Keeping only the first Ball's Axiom (among three) as a definition of truncated Riesz spaces, the first named author and El Adeb proved that if $E$ is truncated Riesz space then $E\oplus\mathbb{R}$ can be equipped with a non-standard structure of Riesz space such that $E$ becomes a Riesz subspace of $E\oplus\mathbb{R}$ and the truncation of $E$ is provided by meet with $1$. In the present paper, we assume that the truncated Riesz space $E$ has a lattice norm $\left\Vert .\right\Vert $ and we give a necessary and sufficient condition for $E\oplus\mathbb{R}$ to have a lattice norm extending $\left\Vert .\right\Vert $. Moreover, we show that under this condition, the set of all lattice norms on $E\oplus\mathbb{R}$ extending $\left\Vert .\right\Vert $ has essentially a largest element $\left\Vert .\right\Vert _{1}$ and a smallest element $\left\Vert .\right\Vert _{0}$. Also, it turns out that any alternative lattice norm on $E\oplus\mathbb{R}$ is either equivalent to $\left\Vert .\right\Vert _{1}$ or equals $\left\Vert .\right\Vert _{0}$. As consequences, we show that $E\oplus\mathbb{R}$ is a Banach lattice if and only if $E$ is a Banach lattice and we get a representation's theorem sustained by the celebrate Kakutani's Representation Theorem.